chapter 2 0 mechanic jan11

8/17/2019 Chapter 2 0 Mechanic Jan11

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CHAPTER 2

Mechanics1


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OBJECTIVE

Coplanar forces in static equilibrium

Distance, time, velocity and acceleration

Projectile motions

Basic Newton’s law Centripetal forces

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SCALARS & VECTORS

Scalar quantity – has only magnitude and is completelyspecified by a number and a unit (eg. Mass, volume,frequency)

Vector quantity – has both magnitude and direction

(eg. Displacement, velocity, force, voltage, current) Vector is represented by an arrow whose length is

proportional to a certain vector quantity and whosedirection indicates the direction of the quantity

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EXAMPLE

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10m, to east

20m, 400

North of East

400

15m, 350

south of west

350


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EXAMPLE

1

2

3

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FORCE - EXAMPLE

Find their resultant force for the figure below:

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80N

100N110N

160N

30o

20o

45o


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CONT..

Method 1: graphically Method 2: rectangular component method

Magnitude X-component Y-component

80 80 cos 0 = 80 80 sin 0 = 0

100 100 cos 45 = 70.71 100 sin 45 = 70.71

110 -110 cos 30 = -95.26 110 sin 30 = 55

160 -160 cos 20 = -150.35 -160 sin 20 = - 54.72

total -94 71

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CONT..

Resultant force, R total = √ (x2 + y2) = √ (-94)2 +(71)2 = 117.80 N

Resultant angle = tan-1 (y/x)

= 2nd

quadrant (from the coordinate)= 143o

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Sum of vector (sum of force) = 117.8N, 1430


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CONT..

Method 3: polar form to

rectangular form

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EXAMPLE

A woman walks eastward for 5 km and then

northward for 10 km.

How far is she from her starting point? (distance)

If she had walked directly to her direction, in what

direction would she have headed? (displacement)

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N

E

S

W


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EXAMPLE

Let F have a magnitude of 300N and make angle,

θ=30o with the positive x direction. Find Fx and Fy

If F=300N and θ=145o (2nd Quadrant), find Fx andFy

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EXAMPLE

The man in figure below exerts a force of 100

N on the wagon at an angle of 30 degrees

above horizontal. Find the horizontal and

vertical components of this force

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VECTOR ADDITION

Components

Simply add together the x-components, y-

components and z-components separately and the

sums are now the x, y and z components of the

resultant.

A = Ax + Ay + Az

B = Bx + By + Bz

C = Cx + Cy + Cz

Resultant, R = Rx +Ry +Rz

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VECTOR ADDITION

where Rx = Ax +Bx +Cx

Ry = Ay +By +Cy

Rz = Az +Bz +Cz

To avoid confusion unit vectors are introduced.

A = Ax i + Ay j + Az k

B = Bx i + By j + Bz k

R = A + B = ( Ax +Bx ) i + ( Ay +By ) j +

( Az + Bz ) k15


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SUBTRACTION OF VECTORS

Using Components

A = Ax i + Ay j + Az k

B = Bx i + By j + Bz k

R = A - B = ( Ax - Bx ) i + ( Ay - By) j + (Az -

Bz) k

Subtract the coefficients of i, j and k separately.

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MULTIPLICATION BY A SCALAR

If a vector R is multiplied by a scalar k the direction

of the vector remains unaltered but the magnitude

is now k R. The resulting vector is kR.

The magnitude of each component is multiplied by

k.

Example: If R = 2i + 3 j + 5k

Then: 2R = 4i + 6 j + 10k

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VECTOR PRODUCT ( "CROSS PRODUCT")

The vector or cross-product C of 2 vectors A B

has:

A

B = i j k Ax Ay Az

Bx By Bz

R= i ( AyBz - AzBy) - j ( AxBz - AzBx ) + k ( AxBy- AyBx )

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MOTION IN ONE DIMENSION

Motion continuous change in the position of an

object

3 types of motion

Translational: car moving down a highway

Rotational: earth’s spin on its axis

Vibrational: back-and-forth movement of a pendulum

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DISTANCE, D

Is a scalar quantity

Is a path length transverse in moving from 1

location to another

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DISPLACEMENT, S

In the study of translational motion, moving

object can be viewed as a particle

regardless of its size

Displacement is a vector quantityDisplacement is defined as distance or the

change in particle’s position,

Ds=sf –si where si initial positions

sf are and final positions21


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EXAMPLE

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Initial position is 0m


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SPEED, V

Scalar quantity

defined as the total distance traveled

divided by the total time it takes to travel

that distance Instantaneous speed, V= distance /t

Average speed is the rate of change of

distance

V = Δd / Δt

= change of distance / time interval24


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VELOCITY, V

Vector quantity Instantaneous velocity, V = s/t

(S refer to the displacement on constant period)

Average velocity is the rate of change ofdisplacement

V = Δs / Δt= change of displacement/time interval

Its direction is in the direction of the

displacementObject moving in uniform velocity if

ds/dt=constant25


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ACCELERATION, A

Vector quantity

Instantaneous acceleration, a = v/t

Average acceleration is the rate of change of

velocitya = Δv / Δt

= change of velocity / time interval

Its direction is in the direction of motion

Acceleration is uniform when magnitude ofvelocity change (dv/dt) at constant rate and fixdirection

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GRAPH REPRESENTATION

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LINEAR MOTION WITH CONSTANT

ACCELERATION

Uniform acceleration, a = (v – v0) / t

v = v0 + at

Displacement, s =1/2 (v0 + v)t

= area of v vs t graph s = v0t +1/2 at

2

v2 = v02 + 2as

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v0 = u = initial velocity

V = final velocity

a = acceleration

T= time

S= displacement


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FREE FALL

Vertical motion with constant acceleration, g undergravitational field without air resistance

g=9.81m/s2; direction toward center of the earth(downward)

a=-g v = v0 – gt v2 = v0

2 - 2gs

s = v0t -1/2 gt2

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*Note:V=0 when particle reach max height

If the free fall just show onedirection, we can assumea=g=9.81m/s2 for easier calculation


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PROJECTILE MOTIONS

Consists of two motions

Vertical component (y-comp): motion under constant

acceleration, -g

Horizontal component (x-comp): motion under constant

velocity

Path followed by a projectile is called trajectory

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CONT..

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CONT..

Initial condition (point A)

X-comp: Vox= Vocos θ constant all time

Y-comp: Voy= Vosin θ varies

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CONT..

Point B and point D X-comp: V1x= V2x=Vox = Vocos θ

(Point B) Y-comp: V1y= Voy – gt1

(Point D) Y-comp: V2y= Voy – gt2

Velocity: Magnitude at (B), V1= √ (V1x)2 + (V1y)2

Direction, θ1= tan-1 (V1y / V1x)

Displacement (S):

X = V0x t

Y = V0y t - ½ gt2

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CONT..

Point C X-comp: Vx= Vox = Vocos θ Y-comp: Vy= 0

Displacement (S):

V2 = u2 + 2as Vy2 = v0y2 - 2gsy 0 = (V0sin θ)2 – 2gH

H= (V02 sin2 θ) / 2g V= u +at

vy = v0y

– gt t = (V0sin θ)/ g

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EXAMPLE

A projectile is fired with an initial velocity of 80 m/s

at an angle of 30oabove the horizontal. Find:

Its position and velocity after 6s

Time required to reach the maximum height

Maximum height

The horizontal range, r (before it strike on floor)

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FORCES

Force is something capable of changing an object’sstate of motion

4 types:

Gravitational force: involve attraction between massive

bodies, long-range force, weakest force in nature

Electromagnetic force: attraction and repulsive force

between electric charges, long-range force

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CONT..

Strong nuclear force: attraction force bonds theneutrons and protons together in nucleus, short-rangeforce, strongest force in nature

Weak nuclear force: cause unstable condition foratomic nucleus and for radioactive decay, short-range

force, 12 time weaker than electromagnetic force Force is vector quantity

F= ma ; kgms-2 / N

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NEWTON’S SECOND LAW

aF m=∑

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The rates of change of linear momentum of amoving body is proportional to the resultantforce and is in the same direction as force acton it

States that the acceleration, a of an object inthe direction of a resultant force, F is directlyproportional to the magnitude of the force andinversely proportional to the mass

1N = 1kgms-2

And the force of gravity or weight,

gFg m


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NEWTON’S THIRD LAW

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Every action must produce an equal and

opposite reaction (not act on same object)

Or whenever one object exerts a forces, F12 on

a second object, the second exerts an equaland opposite force, F21 of the first object

F12 = – F21Someone climb a ladder

Rung must have same but opposite force on the

foot to avoid collapse

Ffoot = - Frung


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EQUILIBRIUM

Occur when the resultant of all external

force is zero

A body in equilibrium must be either at rest

or in motion with constant velocity∑Fx = 0

∑Fy = 0

A body is in transitional equilibrium if andonly if the vector sum of the forces acting on

it is zero.42


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FREE-BODY DIAGRAM

Is a diagram that drawn with all known quantities

are labeled. Then a force diagram indicating all

forces and their components is constructed.

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Circular Motion

Motion along the perimeter of a circle


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Definition of Angular Displacement

1) When a rigid body rotates about a fixed axis, the angulardisplacement is the angle Δθ swept out by a line passing

through any point on the body and intersecting the axis ofrotation perpendicularly.

2) By convention, the angular displacement is positive if it is

counterclockwise and negative if it is clockwise.

SI unit o f Angular Displacement : radian (rad) *


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Angular Displacement

• Angular displacement if often expressed in one ofthree units. The first is the familiar degree, and itis well known that there are 360°in a circle.

•The second unit is the revolution (rev), onerevolution representing one complete turn of 360°.

•The most useful unit from a scientific viewpoint,

however, is the SI unit called the radian (rad).


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Angular Displacement

As the disc rotates, the point traces out and arc of length (s),which is measured along a circle of radius (r). The angle θ will be

in radians:-


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Example 1

Synchronous or “stationary” communication

satellites are put into an orbit whose radius is r =

4.23 x 107 m. The orbit is in the plane of the

equator, and two adjacent satellites have anangular separation of θ = 2.0°. Find the arc length

(s) that separates the satellites.

Solut ion:

Step 1 – Convert degree into radians,

2.0°= (2.0 degrees) (2π radians/ 360 degrees) = 0.0349

radians

Step 2 – Calculation of the arc length,

S = r θ = (4.23 x 107

m) (0.0349 rad) = 1.48 x 106

m


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Angular Velocity

Example 2

A gymnast on a high bar swings through two revolutions

(clockwise) in a time of 1.90 s. Find the average angular

velocity (rad/s) of the gymnast.

Solut ion:

Δθ = -2.0 revolutions (2π radians / 1 revolution)= -12.6 radians

Where the minus sign denotes that the gymnast

rotates clockwise. The average angular velocity is:-

ω= (Δθ / Δt) = (-12.6 rad / 1.90 s) = - 6.63 rad/s


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Definition of Angular Acceleration

In linear motion, a changing velocity means that an

acceleration is occurring. Changing angular velocity means

that an angular acceleration is occuring. The angular

acceleration is defined as :-

The SI unit for angular acceleration is rad/s2


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Angular Acceleration

Example 3

A jet awaiting for takeoff is momentarily stopped on the

runway. As seen from the front of one engine, the fan blades

are rotating with an angular velocity of -110 rad/s, where the

negative sign indicates a clockwise rotation. As the plane

takes off, the angular velocity of the blades reaches – 330

rad/s in a time of 14 s. Find the angular acceleration,

assuming it to be constant.

Solut ion:

α = (ω-ω0)/(t-t0) = (-330 rad/s) – (-110rad/s) / (14 s)

= -16 rad/s2


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The Equations of Rotational Kinematics

Centripetal Acceleration and Tangential Acceleration


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Centripetal Acceleration and Tangential Acceleration

1) When an object picks up speed as it moves around a circle, it

has a tangential acceleration.

2) In addition, the object also has a centripetal acceleration.3) Even when the magnitude of the tangential velocity is

constant, an acceleration is present, since the direction of the

velocity changes continually.

4) Because the resulting acceleration points toward the center of

the circle, it is called CENTRIPETAL acceleration.

ac= VT2 / r

Subscript T is the reminder

for tangential speed.

The centripetal acceleration can

be expressed in terms of angular speed ω:-

ac= (r ω)2 / r

= r ω2


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UNIFORM CIRCULAR MOTION

Movement of object in a circular path atuniform speed

∑Fc = mac

Angular velocity, W (rad/s)Tangential/linear velocity, V = rW

Angular acceleration, α = (Wf – Wi) /tac = centripetal acceleration

= V2 / r = r W2

Centripetal force, F= m(V2/r) = mW2r

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**1 rev = 2π rad


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REFERENCES

Hugh D. Young et al., University

Physics: With Modern Physics, 13th

Edition, 2012.

Giancoli, D.C. Physics for Scientistsand Engineers with Modern Physics.

4th Edition. Pearson, 2009.

Knight, R.D., Jones, B., Field, S.,College Physics. 2nd Edition.

Pearson/Addison Wesley, 2010.

chapter 2 0 mechanic jan11

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